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Cell["Exercises lesson 4", "Title",
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Cell["\<\
1. ADAPTIVE DYNAMICS / GAME THEORY
Doebeli et al. (2004, Science) discuss an evolutionary game whereby \
individuals can make a continuous investment in a cooperative trait that can \
go between 0 and 1 and in which the payoff function of interacting with \
someone else is given by
payoff(y,z)=-c1*y-c2*y^2+b1*(y+z)+b2*(y+z)^2   where y is the individual's \
level of cooperation and z is the social interactant's level of cooperation

Calculate invasion fitness, make a pairwise invasibility plot, identify \
evolutionarily singular strategies and assess stability criteria for the case \
where
b2 = -1.4; b1 = 6; c2 = -1.6; c1 = 4.56;\
\>", "Subsection",
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Cell["Invasion fitness is given by:", "Text",
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\>"]]
}, Open  ]],

Cell["\<\
The selection differential for an increased investment in cooperation (y) is \
given by\
\>", "Text",
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Cell[BoxData[
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Cell["\<\
And an evolutionarily singular strategy occurs when individuals invest in a \
level of cooperation of z*=0.6 :\
\>", "Text",
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Cell["\<\
The following two second-order derivatives determine the stability and \
attainability of this strategy:\
\>", "Text",
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Cell["\<\
The evolutionarily singular strategy is NOT evolutionarily stable since B > 0 \
(ie not immune to invasion by neighbouring types)

But the strategy IS convergence stable (attracting) since A > B

This means that the identified singular strategy is an evolutionary branching \
point.

The strategy is also capable of invading into all of its neighbouring types \
since A > 0 (invasion potential)\
\>", "Text",
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Cell["\<\
What happens once the branching point is reached can be readily simulated on \
a computer. You can see that the result is that the population splits up into \
two pure types of cooperators and defectors:\
\>", "Text",
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Cell["\<\
2. INCLUSIVE FITNESS
Last week we used a genetic model to show that in social Hymenoptera the \
optimum sex-ratio from the perspective of the mother queen is 1:1 whereas \
from the perspective from the workers it is 3:1 (under single mating, or 2:1 \
under double mating). Now try to derive the same result using inclusive \
fitness methods, whereby you write the relative production of new queens and \
males by a focal colony that produces a proportion f new queens in a \
population where colonies produce a proportion F queens. Calculate the \
inclusive fitness effect by looking at the effects of increasing f on the \
relative production of queens and males, wheighed by relatedness to queens \
and males (where relatedness is calculated either from the perspective of the \
mother queen or from the perspective of the workers).\
\>", "Subsection",
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Cell["\<\
Assume that a focal colony produces a proportion f new queens and 1-f males \
in a population where colonies produce a proportion F queens.
The relative production of new queens and males reared by this colony is then \
given by\
\>", "Text",
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Cell["\<\
The inclusive fitness effect of increased rearing of new queens is given by\
\>", "Text",
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Cell["\<\
where Rf and Rm are the life-for-life relatedness to queens and males reared \
in the colony (= product of regression relatedness and reproductive value).\
\>", "Text",
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Cell["\<\
An ESS is reached when this inclusive fitness effect becomes zero, which \
occurs when a proportion of females f*=Rf/(Rf+Rm) is reared:\
\>", "Text",
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Cell[CellGroupData[{

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Cell["\<\
From the perspective of the mother queen the relatedness to new queens and \
males reared is both 1/2. Hence, from her perspective she favours an equal \
sex ratio (f*=1/2):\
\>", "Text",
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Cell["\<\
Workers, however, as a result of haplodiploidy are three times more related \
to sisters (3/4) than to brother (1/4). Hence, from their perspective they \
favour a 3:1 sex ratio (f*=3/4):\
\>", "Text",
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Cell["\<\
And this is the workers' optimum under double mating, where Rf=1/2 and \
Rm=1/4: a 2:1 sex ratio (f*=2/3):\
\>", "Text",
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Cell[CellGroupData[{

Cell["\<\
3. INCLUSIVE FITNESS - Calculation of reproductive value
Calculate the relative reproductive value of males and females in \
haplodiploid social Hymenoptera assuming that the males are all produced by \
the workers rather than by the mother queen \
\>", "Subsection",
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Cell["\<\
Our recurrence equations now just become:
frequency of allele in queens in next generation pf\[CloseCurlyQuote] = \
(1/2).pf + (1/2).pm
frequency of allele in males in next generation pm\[CloseCurlyQuote] = \
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If we put the gene transmission probabilities in a matrix A we get:\
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The individual reproductive values rate of increase of our gene is given by \
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And our stable class frequences and individual reproductive values are given \
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I.e. when workers produce all the males females and males have equal \
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